Thread: Sequence proof

If you have a sequence a_n which is bounded below and a sequence b_n which is bounded below, show that a_n + b_n is bounded below. The definition of bounded below is a_n > -M.

Originally Posted by natester

If you have a sequence a_n which is bounded below and a sequence b_n which is bounded below, show that a_n + b_n is bounded below. The definition of bounded below is a_n > -M.

your definition of being bounded below seems to be lacking. anyway, if that's what you are given, here is how we proceed. (i will write M instead of -M, 'cause that looks weird).

so we have and for all .

thus, for all n. so is bounded below by M + K

That makes since to me, but how would I prove that the statement a_n + b_n > M + K is true?

Originally Posted by natester

That makes since to me, but how would I prove that the statement a_n + b_n > M + K is true?

um, it's not really something you have to prove. unless you want to go down to some kind of axiomatic level of number theory, which i doubt. it is just common sense that if you diminish each term, the result will be smaller

like since 2 > 1 and 3 > 1 it just makes sense that 2 + 3 > 1 + 1, because in place of two larger numbers, we put two smaller numbers, so the result must be smaller ...